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<FONT color="green">001</FONT>    /*<a name="line.1"></a>
<FONT color="green">002</FONT>     * Licensed to the Apache Software Foundation (ASF) under one or more<a name="line.2"></a>
<FONT color="green">003</FONT>     * contributor license agreements.  See the NOTICE file distributed with<a name="line.3"></a>
<FONT color="green">004</FONT>     * this work for additional information regarding copyright ownership.<a name="line.4"></a>
<FONT color="green">005</FONT>     * The ASF licenses this file to You under the Apache License, Version 2.0<a name="line.5"></a>
<FONT color="green">006</FONT>     * (the "License"); you may not use this file except in compliance with<a name="line.6"></a>
<FONT color="green">007</FONT>     * the License.  You may obtain a copy of the License at<a name="line.7"></a>
<FONT color="green">008</FONT>     *<a name="line.8"></a>
<FONT color="green">009</FONT>     *      http://www.apache.org/licenses/LICENSE-2.0<a name="line.9"></a>
<FONT color="green">010</FONT>     *<a name="line.10"></a>
<FONT color="green">011</FONT>     * Unless required by applicable law or agreed to in writing, software<a name="line.11"></a>
<FONT color="green">012</FONT>     * distributed under the License is distributed on an "AS IS" BASIS,<a name="line.12"></a>
<FONT color="green">013</FONT>     * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.<a name="line.13"></a>
<FONT color="green">014</FONT>     * See the License for the specific language governing permissions and<a name="line.14"></a>
<FONT color="green">015</FONT>     * limitations under the License.<a name="line.15"></a>
<FONT color="green">016</FONT>     */<a name="line.16"></a>
<FONT color="green">017</FONT>    <a name="line.17"></a>
<FONT color="green">018</FONT>    package org.apache.commons.math3.analysis.function;<a name="line.18"></a>
<FONT color="green">019</FONT>    <a name="line.19"></a>
<FONT color="green">020</FONT>    import org.apache.commons.math3.analysis.DifferentiableUnivariateFunction;<a name="line.20"></a>
<FONT color="green">021</FONT>    import org.apache.commons.math3.analysis.FunctionUtils;<a name="line.21"></a>
<FONT color="green">022</FONT>    import org.apache.commons.math3.analysis.UnivariateFunction;<a name="line.22"></a>
<FONT color="green">023</FONT>    import org.apache.commons.math3.analysis.differentiation.DerivativeStructure;<a name="line.23"></a>
<FONT color="green">024</FONT>    import org.apache.commons.math3.analysis.differentiation.UnivariateDifferentiableFunction;<a name="line.24"></a>
<FONT color="green">025</FONT>    import org.apache.commons.math3.util.FastMath;<a name="line.25"></a>
<FONT color="green">026</FONT>    <a name="line.26"></a>
<FONT color="green">027</FONT>    /**<a name="line.27"></a>
<FONT color="green">028</FONT>     * &lt;a href="http://en.wikipedia.org/wiki/Sinc_function"&gt;Sinc&lt;/a&gt; function,<a name="line.28"></a>
<FONT color="green">029</FONT>     * defined by<a name="line.29"></a>
<FONT color="green">030</FONT>     * &lt;pre&gt;&lt;code&gt;<a name="line.30"></a>
<FONT color="green">031</FONT>     *   sinc(x) = 1            if x = 0,<a name="line.31"></a>
<FONT color="green">032</FONT>     *             sin(x) / x   otherwise.<a name="line.32"></a>
<FONT color="green">033</FONT>     * &lt;/code&gt;&lt;/pre&gt;<a name="line.33"></a>
<FONT color="green">034</FONT>     *<a name="line.34"></a>
<FONT color="green">035</FONT>     * @since 3.0<a name="line.35"></a>
<FONT color="green">036</FONT>     * @version $Id: Sinc.java 1383441 2012-09-11 14:56:39Z luc $<a name="line.36"></a>
<FONT color="green">037</FONT>     */<a name="line.37"></a>
<FONT color="green">038</FONT>    public class Sinc implements UnivariateDifferentiableFunction, DifferentiableUnivariateFunction {<a name="line.38"></a>
<FONT color="green">039</FONT>        /**<a name="line.39"></a>
<FONT color="green">040</FONT>         * Value below which the computations are done using Taylor series.<a name="line.40"></a>
<FONT color="green">041</FONT>         * &lt;p&gt;<a name="line.41"></a>
<FONT color="green">042</FONT>         * The Taylor series for sinc even order derivatives are:<a name="line.42"></a>
<FONT color="green">043</FONT>         * &lt;pre&gt;<a name="line.43"></a>
<FONT color="green">044</FONT>         * d^(2n)sinc/dx^(2n)     = Sum_(k&gt;=0) (-1)^(n+k) / ((2k)!(2n+2k+1)) x^(2k)<a name="line.44"></a>
<FONT color="green">045</FONT>         *                        = (-1)^n     [ 1/(2n+1) - x^2/(4n+6) + x^4/(48n+120) - x^6/(1440n+5040) + O(x^8) ]<a name="line.45"></a>
<FONT color="green">046</FONT>         * &lt;/pre&gt;<a name="line.46"></a>
<FONT color="green">047</FONT>         * &lt;/p&gt;<a name="line.47"></a>
<FONT color="green">048</FONT>         * &lt;p&gt;<a name="line.48"></a>
<FONT color="green">049</FONT>         * The Taylor series for sinc odd order derivatives are:<a name="line.49"></a>
<FONT color="green">050</FONT>         * &lt;pre&gt;<a name="line.50"></a>
<FONT color="green">051</FONT>         * d^(2n+1)sinc/dx^(2n+1) = Sum_(k&gt;=0) (-1)^(n+k+1) / ((2k+1)!(2n+2k+3)) x^(2k+1)<a name="line.51"></a>
<FONT color="green">052</FONT>         *                        = (-1)^(n+1) [ x/(2n+3) - x^3/(12n+30) + x^5/(240n+840) - x^7/(10080n+45360) + O(x^9) ]<a name="line.52"></a>
<FONT color="green">053</FONT>         * &lt;/pre&gt;<a name="line.53"></a>
<FONT color="green">054</FONT>         * &lt;/p&gt;<a name="line.54"></a>
<FONT color="green">055</FONT>         * &lt;p&gt;<a name="line.55"></a>
<FONT color="green">056</FONT>         * So the ratio of the fourth term with respect to the first term<a name="line.56"></a>
<FONT color="green">057</FONT>         * is always smaller than x^6/720, for all derivative orders.<a name="line.57"></a>
<FONT color="green">058</FONT>         * This implies that neglecting this term and using only the first three terms induces<a name="line.58"></a>
<FONT color="green">059</FONT>         * a relative error bounded by x^6/720. The SHORTCUT value is chosen such that this<a name="line.59"></a>
<FONT color="green">060</FONT>         * relative error is below double precision accuracy when |x| &lt;= SHORTCUT.<a name="line.60"></a>
<FONT color="green">061</FONT>         * &lt;/p&gt;<a name="line.61"></a>
<FONT color="green">062</FONT>         */<a name="line.62"></a>
<FONT color="green">063</FONT>        private static final double SHORTCUT = 6.0e-3;<a name="line.63"></a>
<FONT color="green">064</FONT>        /** For normalized sinc function. */<a name="line.64"></a>
<FONT color="green">065</FONT>        private final boolean normalized;<a name="line.65"></a>
<FONT color="green">066</FONT>    <a name="line.66"></a>
<FONT color="green">067</FONT>        /**<a name="line.67"></a>
<FONT color="green">068</FONT>         * The sinc function, {@code sin(x) / x}.<a name="line.68"></a>
<FONT color="green">069</FONT>         */<a name="line.69"></a>
<FONT color="green">070</FONT>        public Sinc() {<a name="line.70"></a>
<FONT color="green">071</FONT>            this(false);<a name="line.71"></a>
<FONT color="green">072</FONT>        }<a name="line.72"></a>
<FONT color="green">073</FONT>    <a name="line.73"></a>
<FONT color="green">074</FONT>        /**<a name="line.74"></a>
<FONT color="green">075</FONT>         * Instantiates the sinc function.<a name="line.75"></a>
<FONT color="green">076</FONT>         *<a name="line.76"></a>
<FONT color="green">077</FONT>         * @param normalized If {@code true}, the function is<a name="line.77"></a>
<FONT color="green">078</FONT>         * &lt;code&gt; sin(&amp;pi;x) / &amp;pi;x&lt;/code&gt;, otherwise {@code sin(x) / x}.<a name="line.78"></a>
<FONT color="green">079</FONT>         */<a name="line.79"></a>
<FONT color="green">080</FONT>        public Sinc(boolean normalized) {<a name="line.80"></a>
<FONT color="green">081</FONT>            this.normalized = normalized;<a name="line.81"></a>
<FONT color="green">082</FONT>        }<a name="line.82"></a>
<FONT color="green">083</FONT>    <a name="line.83"></a>
<FONT color="green">084</FONT>        /** {@inheritDoc} */<a name="line.84"></a>
<FONT color="green">085</FONT>        public double value(final double x) {<a name="line.85"></a>
<FONT color="green">086</FONT>            final double scaledX = normalized ? FastMath.PI * x : x;<a name="line.86"></a>
<FONT color="green">087</FONT>            if (FastMath.abs(scaledX) &lt;= SHORTCUT) {<a name="line.87"></a>
<FONT color="green">088</FONT>                // use Taylor series<a name="line.88"></a>
<FONT color="green">089</FONT>                final double scaledX2 = scaledX * scaledX;<a name="line.89"></a>
<FONT color="green">090</FONT>                return ((scaledX2 - 20) * scaledX2 + 120) / 120;<a name="line.90"></a>
<FONT color="green">091</FONT>            } else {<a name="line.91"></a>
<FONT color="green">092</FONT>                // use definition expression<a name="line.92"></a>
<FONT color="green">093</FONT>                return FastMath.sin(scaledX) / scaledX;<a name="line.93"></a>
<FONT color="green">094</FONT>            }<a name="line.94"></a>
<FONT color="green">095</FONT>        }<a name="line.95"></a>
<FONT color="green">096</FONT>    <a name="line.96"></a>
<FONT color="green">097</FONT>        /** {@inheritDoc}<a name="line.97"></a>
<FONT color="green">098</FONT>         * @deprecated as of 3.1, replaced by {@link #value(DerivativeStructure)}<a name="line.98"></a>
<FONT color="green">099</FONT>         */<a name="line.99"></a>
<FONT color="green">100</FONT>        @Deprecated<a name="line.100"></a>
<FONT color="green">101</FONT>        public UnivariateFunction derivative() {<a name="line.101"></a>
<FONT color="green">102</FONT>            return FunctionUtils.toDifferentiableUnivariateFunction(this).derivative();<a name="line.102"></a>
<FONT color="green">103</FONT>        }<a name="line.103"></a>
<FONT color="green">104</FONT>    <a name="line.104"></a>
<FONT color="green">105</FONT>        /** {@inheritDoc}<a name="line.105"></a>
<FONT color="green">106</FONT>         * @since 3.1<a name="line.106"></a>
<FONT color="green">107</FONT>         */<a name="line.107"></a>
<FONT color="green">108</FONT>        public DerivativeStructure value(final DerivativeStructure t) {<a name="line.108"></a>
<FONT color="green">109</FONT>    <a name="line.109"></a>
<FONT color="green">110</FONT>            final double scaledX  = (normalized ? FastMath.PI : 1) * t.getValue();<a name="line.110"></a>
<FONT color="green">111</FONT>            final double scaledX2 = scaledX * scaledX;<a name="line.111"></a>
<FONT color="green">112</FONT>    <a name="line.112"></a>
<FONT color="green">113</FONT>            double[] f = new double[t.getOrder() + 1];<a name="line.113"></a>
<FONT color="green">114</FONT>    <a name="line.114"></a>
<FONT color="green">115</FONT>            if (FastMath.abs(scaledX) &lt;= SHORTCUT) {<a name="line.115"></a>
<FONT color="green">116</FONT>    <a name="line.116"></a>
<FONT color="green">117</FONT>                for (int i = 0; i &lt; f.length; ++i) {<a name="line.117"></a>
<FONT color="green">118</FONT>                    final int k = i / 2;<a name="line.118"></a>
<FONT color="green">119</FONT>                    if ((i &amp; 0x1) == 0) {<a name="line.119"></a>
<FONT color="green">120</FONT>                        // even derivation order<a name="line.120"></a>
<FONT color="green">121</FONT>                        f[i] = (((k &amp; 0x1) == 0) ? 1 : -1) *<a name="line.121"></a>
<FONT color="green">122</FONT>                               (1.0 / (i + 1) - scaledX2 * (1.0 / (2 * i + 6) - scaledX2 / (24 * i + 120)));<a name="line.122"></a>
<FONT color="green">123</FONT>                    } else {<a name="line.123"></a>
<FONT color="green">124</FONT>                        // odd derivation order<a name="line.124"></a>
<FONT color="green">125</FONT>                        f[i] = (((k &amp; 0x1) == 0) ? -scaledX : scaledX) *<a name="line.125"></a>
<FONT color="green">126</FONT>                               (1.0 / (i + 2) - scaledX2 * (1.0 / (6 * i + 24) - scaledX2 / (120 * i + 720)));<a name="line.126"></a>
<FONT color="green">127</FONT>                    }<a name="line.127"></a>
<FONT color="green">128</FONT>                }<a name="line.128"></a>
<FONT color="green">129</FONT>    <a name="line.129"></a>
<FONT color="green">130</FONT>            } else {<a name="line.130"></a>
<FONT color="green">131</FONT>    <a name="line.131"></a>
<FONT color="green">132</FONT>                final double inv = 1 / scaledX;<a name="line.132"></a>
<FONT color="green">133</FONT>                final double cos = FastMath.cos(scaledX);<a name="line.133"></a>
<FONT color="green">134</FONT>                final double sin = FastMath.sin(scaledX);<a name="line.134"></a>
<FONT color="green">135</FONT>    <a name="line.135"></a>
<FONT color="green">136</FONT>                f[0] = inv * sin;<a name="line.136"></a>
<FONT color="green">137</FONT>    <a name="line.137"></a>
<FONT color="green">138</FONT>                // the nth order derivative of sinc has the form:<a name="line.138"></a>
<FONT color="green">139</FONT>                // dn(sinc(x)/dxn = [S_n(x) sin(x) + C_n(x) cos(x)] / x^(n+1)<a name="line.139"></a>
<FONT color="green">140</FONT>                // where S_n(x) is an even polynomial with degree n-1 or n (depending on parity)<a name="line.140"></a>
<FONT color="green">141</FONT>                // and C_n(x) is an odd polynomial with degree n-1 or n (depending on parity)<a name="line.141"></a>
<FONT color="green">142</FONT>                // S_0(x) = 1, S_1(x) = -1, S_2(x) = -x^2 + 2, S_3(x) = 3x^2 - 6...<a name="line.142"></a>
<FONT color="green">143</FONT>                // C_0(x) = 0, C_1(x) = x, C_2(x) = -2x, C_3(x) = -x^3 + 6x...<a name="line.143"></a>
<FONT color="green">144</FONT>                // the general recurrence relations for S_n and C_n are:<a name="line.144"></a>
<FONT color="green">145</FONT>                // S_n(x) = x S_(n-1)'(x) - n S_(n-1)(x) - x C_(n-1)(x)<a name="line.145"></a>
<FONT color="green">146</FONT>                // C_n(x) = x C_(n-1)'(x) - n C_(n-1)(x) + x S_(n-1)(x)<a name="line.146"></a>
<FONT color="green">147</FONT>                // as per polynomials parity, we can store both S_n and C_n in the same array<a name="line.147"></a>
<FONT color="green">148</FONT>                final double[] sc = new double[f.length];<a name="line.148"></a>
<FONT color="green">149</FONT>                sc[0] = 1;<a name="line.149"></a>
<FONT color="green">150</FONT>    <a name="line.150"></a>
<FONT color="green">151</FONT>                double coeff = inv;<a name="line.151"></a>
<FONT color="green">152</FONT>                for (int n = 1; n &lt; f.length; ++n) {<a name="line.152"></a>
<FONT color="green">153</FONT>    <a name="line.153"></a>
<FONT color="green">154</FONT>                    double s = 0;<a name="line.154"></a>
<FONT color="green">155</FONT>                    double c = 0;<a name="line.155"></a>
<FONT color="green">156</FONT>    <a name="line.156"></a>
<FONT color="green">157</FONT>                    // update and evaluate polynomials S_n(x) and C_n(x)<a name="line.157"></a>
<FONT color="green">158</FONT>                    final int kStart;<a name="line.158"></a>
<FONT color="green">159</FONT>                    if ((n &amp; 0x1) == 0) {<a name="line.159"></a>
<FONT color="green">160</FONT>                        // even derivation order, S_n is degree n and C_n is degree n-1<a name="line.160"></a>
<FONT color="green">161</FONT>                        sc[n] = 0;<a name="line.161"></a>
<FONT color="green">162</FONT>                        kStart = n;<a name="line.162"></a>
<FONT color="green">163</FONT>                    } else {<a name="line.163"></a>
<FONT color="green">164</FONT>                        // odd derivation order, S_n is degree n-1 and C_n is degree n<a name="line.164"></a>
<FONT color="green">165</FONT>                        sc[n] = sc[n - 1];<a name="line.165"></a>
<FONT color="green">166</FONT>                        c = sc[n];<a name="line.166"></a>
<FONT color="green">167</FONT>                        kStart = n - 1;<a name="line.167"></a>
<FONT color="green">168</FONT>                    }<a name="line.168"></a>
<FONT color="green">169</FONT>    <a name="line.169"></a>
<FONT color="green">170</FONT>                    // in this loop, k is always even<a name="line.170"></a>
<FONT color="green">171</FONT>                    for (int k = kStart; k &gt; 1; k -= 2) {<a name="line.171"></a>
<FONT color="green">172</FONT>    <a name="line.172"></a>
<FONT color="green">173</FONT>                        // sine part<a name="line.173"></a>
<FONT color="green">174</FONT>                        sc[k]     = (k - n) * sc[k] - sc[k - 1];<a name="line.174"></a>
<FONT color="green">175</FONT>                        s         = s * scaledX2 + sc[k];<a name="line.175"></a>
<FONT color="green">176</FONT>    <a name="line.176"></a>
<FONT color="green">177</FONT>                        // cosine part<a name="line.177"></a>
<FONT color="green">178</FONT>                        sc[k - 1] = (k - 1 - n) * sc[k - 1] + sc[k -2];<a name="line.178"></a>
<FONT color="green">179</FONT>                        c         = c * scaledX2 + sc[k - 1];<a name="line.179"></a>
<FONT color="green">180</FONT>    <a name="line.180"></a>
<FONT color="green">181</FONT>                    }<a name="line.181"></a>
<FONT color="green">182</FONT>                    sc[0] *= -n;<a name="line.182"></a>
<FONT color="green">183</FONT>                    s      = s * scaledX2 + sc[0];<a name="line.183"></a>
<FONT color="green">184</FONT>    <a name="line.184"></a>
<FONT color="green">185</FONT>                    coeff *= inv;<a name="line.185"></a>
<FONT color="green">186</FONT>                    f[n]   = coeff * (s * sin + c * scaledX * cos);<a name="line.186"></a>
<FONT color="green">187</FONT>    <a name="line.187"></a>
<FONT color="green">188</FONT>                }<a name="line.188"></a>
<FONT color="green">189</FONT>    <a name="line.189"></a>
<FONT color="green">190</FONT>            }<a name="line.190"></a>
<FONT color="green">191</FONT>    <a name="line.191"></a>
<FONT color="green">192</FONT>            if (normalized) {<a name="line.192"></a>
<FONT color="green">193</FONT>                double scale = FastMath.PI;<a name="line.193"></a>
<FONT color="green">194</FONT>                for (int i = 1; i &lt; f.length; ++i) {<a name="line.194"></a>
<FONT color="green">195</FONT>                    f[i]  *= scale;<a name="line.195"></a>
<FONT color="green">196</FONT>                    scale *= FastMath.PI;<a name="line.196"></a>
<FONT color="green">197</FONT>                }<a name="line.197"></a>
<FONT color="green">198</FONT>            }<a name="line.198"></a>
<FONT color="green">199</FONT>    <a name="line.199"></a>
<FONT color="green">200</FONT>            return t.compose(f);<a name="line.200"></a>
<FONT color="green">201</FONT>    <a name="line.201"></a>
<FONT color="green">202</FONT>        }<a name="line.202"></a>
<FONT color="green">203</FONT>    <a name="line.203"></a>
<FONT color="green">204</FONT>    }<a name="line.204"></a>




























































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